# Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes

arXiv: https://arxiv.org/abs/2304.11247 · HTML: https://mlsanity.com/paper/2304.11247/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 72 references that are in the map.
- First posted on arXiv 21 Apr 2023.

## Builds on (most cited first, 27 of 27)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [What is being transferred in transfer learning?](https://mlsanity.com/paper/2008.11687/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Transfer learning in hybrid classical-quantum neural networks](https://mlsanity.com/paper/1912.08278/) (2019)
- [QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer](https://mlsanity.com/paper/2102.12846/) (2021)
- [Quantum machine learning for image classification](https://mlsanity.com/paper/2304.09224/) (2023)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Foundations for Near-Term Quantum Natural Language Processing](https://mlsanity.com/paper/2012.03755/) (2020)
- [Hybrid quantum neural network for drug response prediction](https://mlsanity.com/paper/2211.05777/) (2022)
- [Quantum Deep Hedging](https://mlsanity.com/paper/2303.16585/) (2023)
- [Quantum versus Classical Generative Modelling in Finance](https://mlsanity.com/paper/2008.00691/) (2020)
- [Benchmarking simulated and physical quantum processing units using quantum and hybrid algorithms](https://mlsanity.com/paper/2211.15631/) (2022)
- [Hybrid quantum ResNet for car classification and its hyperparameter optimization](https://mlsanity.com/paper/2205.04878/) (2022)
- [An exponentially-growing family of universal quantum circuits](https://mlsanity.com/paper/2212.00736/) (2022)
- [Quantum algorithms applied to satellite mission planning for Earth observation](https://mlsanity.com/paper/2302.07181/) (2023)
- [Practical application-specific advantage through hybrid quantum computing](https://mlsanity.com/paper/2205.04858/) (2022)
- [Parallel Hybrid Networks: an interplay between quantum and classical neural networks](https://mlsanity.com/paper/2303.03227/) (2023)
- [QDNN: DNN with Quantum Neural Network Layers](https://mlsanity.com/paper/1912.12660/) (2019)
- [Tetra-AML: Automatic Machine Learning via Tensor Networks](https://mlsanity.com/paper/2303.16214/) (2023)
- [QSpeech: Low-Qubit Quantum Speech Application Toolkit](https://mlsanity.com/paper/2205.13221/) (2022)

## Built on by (most cited first, 18 of 18)

- [Let Quantum Neural Networks Choose Their Own Frequencies](https://mlsanity.com/paper/2309.03279/) (2023)
- [Hybrid quantum image classification and federated learning for hepatic steatosis diagnosis](https://mlsanity.com/paper/2311.02402/) (2023)
- [Forecasting steam mass flow in power plants using the parallel hybrid network](https://mlsanity.com/paper/2307.09483/) (2023)
- [QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2503.16678/) (2025)
- [Hybrid quantum cycle generative adversarial network for small molecule generation](https://mlsanity.com/paper/2402.00014/) (2023)
- [A supervised hybrid quantum machine learning solution to the emergency escape routing problem](https://mlsanity.com/paper/2307.15682/) (2023)
- [Photovoltaic power forecasting using quantum machine learning](https://mlsanity.com/paper/2312.16379/) (2023)
- [Method for noise-induced regularization in quantum neural networks](https://mlsanity.com/paper/2410.19921/) (2024)
- [Predictive control of blast furnace temperature in steelmaking with hybrid depth-infused quantum neural networks](https://mlsanity.com/paper/2504.12389/) (2025)
- [A Fast and Generalizable Fourier Neural Operator-Based Surrogate for Melt-Pool Prediction in Laser Processing](https://mlsanity.com/paper/2602.06241/) (2026)
- [Fatigue-PINN: Physics-Informed Fatigue-Driven Motion Modulation and Synthesis](https://mlsanity.com/paper/2502.19056/) (2025)
- [Data-driven quantum Koopman method for simulating nonlinear dynamics](https://mlsanity.com/paper/2507.21890/) (2025)
- [Hybrid Classical-Quantum architecture for vectorised image classification of hand-written sketches](https://mlsanity.com/paper/2407.06416/) (2024)
- [Enhancing Physics-Informed Neural Networks with a Hybrid Parallel Kolmogorov-Arnold and MLP Architecture](https://mlsanity.com/paper/2503.23289/) (2025)
- [Wavelet-Accelerated Physics-Informed Quantum Neural Network for Multiscale Partial Differential Equations](https://mlsanity.com/paper/2512.08256/) (2025)
- [Multi-objective optimization and quantum hybridization of equivariant deep learning interatomic potentials](https://mlsanity.com/paper/2602.16908/) (2026)
- [Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations](https://mlsanity.com/paper/2605.27408/) (2026)
- [Quantum Reservoir Computing with Physics-Informed Correction for Reduced-Order PDE Forecasting](https://mlsanity.com/paper/2608.23119/) (2026)

## Abstract (arXiv)

Finding the distribution of the velocities and pressures of a fluid by solving the Navier-Stokes equations is a principal task in the chemical, energy, and pharmaceutical industries, as well as in mechanical engineering and the design of pipeline systems. With existing solvers, such as OpenFOAM and Ansys, simulations of fluid dynamics in intricate geometries are computationally expensive and require re-simulation whenever the geometric parameters or the initial and boundary conditions are altered. Physics-informed neural networks are a promising tool for simulating fluid flows in complex geometries, as they can adapt to changes in the geometry and mesh definitions, allowing for generalization across fluid parameters and transfer learning across different shapes. We present a hybrid quantum physics-informed neural network that simulates laminar fluid flows in 3D Y-shaped mixers. Our approach combines the expressive power of a quantum model with the flexibility of a physics-informed neural network, resulting in a 21% higher accuracy compared to a purely classical neural network. Our findings highlight the potential of machine learning approaches, and in particular hybrid quantum physics-informed neural network, for complex shape optimization tasks in computational fluid dynamics. By improving the accuracy of fluid simulations in complex geometries, our research using hybrid quantum models contributes to the development of more efficient and reliable fluid dynamics solvers.